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Research Papers

On Microstructure Evolution in Fiber-Reinforced Elastomers and Implications for Their Mechanical Response and Stability

[+] Author and Article Information
Oscar Lopez-Pamies

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300oscar.lopez-pamies@sunysb.edu

Martín I. Idiart

Área Departamental Aeronáutica, Facultad de Ingeniería, Universidad Nacional de La Plata, Calles 1 y 47, La Plata B1900TAG, Argentina; Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Avenida Rivadavia 1917, Ciudad de Buenos Aires C1033AAJ, Argentinamartin.idiart@ing.unlp.edu.ar

Zhiyun Li

Department of Mechanical Engineering, State University of New York, Stony Brook, NY 11794-2300

Here, it is relevant to mention that other definitions of macroscopic behavior for hyperelastic composites have been proposed in the literature, including the notion of a “globally equivalent homogeneous” material (9).

J. Eng. Mater. Technol 133(1), 011007 (Dec 01, 2010) (10 pages) doi:10.1115/1.4002642 History: Received February 17, 2010; Revised April 16, 2010; Published December 01, 2010; Online December 01, 2010

Lopez-Pamies and Idiart (2010, “Fiber-Reinforced Hyperelastic Solids: A Realizable Homogenization Constitutive Theory,” J. Eng. Math., 68(1), pp. 57–83) have recently put forward a homogenization theory with the capability to generate exact results not only for the macroscopic response and stability but also for the evolution of the microstructure in fiber-reinforced hyperelastic solids subjected to finite deformations. In this paper, we make use of this new theory to construct exact, closed-form solutions for the change in size, shape, and orientation undergone by the underlying fibers in a model class of fiber-reinforced hyperelastic solids along arbitrary 3D loading conditions. Making use of these results, we then establish connections between the evolution of the microstructure and the overall stress-strain relation and macroscopic stability in fiber-reinforced elastomers. In particular, we show that the rotation of the fibers may lead to the softening of the overall stiffness of fiber-reinforced elastomers under certain loading conditions. Furthermore, we show that this geometric mechanism is intimately related to the development of long-wavelength instabilities. These findings are discussed in light of comparisons with recent results for related material systems.

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Figures

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Figure 1

Schematic illustrating a fiber-reinforced elastomer in the undeformed (Ω0) and deformed (Ω) configurations; note that for convenience, the initial orientation of the fibers N has been aligned with the coordinate basis vector e3

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Figure 2

Schematic representation of the evolution of microstructure in a fiber-reinforced elastomer along a loading path with macroscopic deformation gradient F¯: (a) In the undeformed configuration, a typical fiber has a circular cross section (i.e., semiaxes z1−2=z2−2=1) and its cylindrical axis (with semiaxis z3−2=∞) is aligned with the N=e3 direction. (b) In the deformed configuration, the orientation of the fibers evolves to v3, the eigenvector associated with the zero eigenvalue (z3) of ZTZ. In addition, the initial circular cross section evolves into an elliptical cross section with semiaxes and principal directions that are characterized by the eigenvalues z1 and z2 and corresponding eigenvectors v1 and v2, of ZTZ (see Eqs. 6,7).

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Figure 3

Pictorial representation of the applied loading conditions 38 and the resulting evolution of the orientation of the fibers as determined by relations 42,43

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Figure 4

Angle of rotation of the fibers φ (given by expression 43) in fiber-reinforced neo-Hookean elastomers subjected to axisymmetric compression (Eq. 38) for various initial fiber orientations φ0 as a function of the applied macroscopic stretch λ¯; Note that the results are completely independent of the constitutive behavior of the matrix and fibers, as well as of the volume fraction of fibers

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Figure 5

Overall constitutive response (see Eq. 40) of fiber-reinforced neo-Hookean elastomers subjected to axisymmetric compression: (a) results for the stiffness dS¯/dλ¯ in the ground state (λ¯=1) for fiber-to-matrix heterogeneity contrasts t=μ(2)/μ(1)=5, 20, 50 and volume fraction of fibers c0=30% as a function of the initial fiber orientation φ0 and (b) results for the overall stress S¯ for φ0=10 deg, 35.3 deg, 60 deg, 80 deg, 90 deg, t=20, and c0=30% as a function of the applied loading λ¯

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Figure 6

Onset of macroscopic instabilities (see Eq. 41) in fiber-reinforced neo-Hookean elastomers subjected to axisymmetric compression: (a) results for the critical deformation λ¯cr at which instabilities may first develop for fiber-to-matrix heterogeneity contrasts t=μ(2)/μ(1)=5, 20, 50 and volume fraction of fibers c0=30% as a function of the initial fiber orientation φ0 and (b) results for λ¯cr for φ0=90 deg and c0=10%, 30%, 50% as a function of t

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